Some classical features of the Links-Gould invariants of links | BIMSA Topology Seminar

Abstract

For a long time, the Alexander polynomial was the main easily computable link invariant to be knownBut in 1984, Jones discovered his well known polynomial link invariant, and that gave birth to the vasitheory of quantum link invariants. However, unlike for the Alexander invariant, it is in general hard todeduce precise topological properties on a knot or link from the value quantum invariants take on thatlink. For instance, no genus bound is known for the Jones polynomial.

The Links-Gould invariants of oriented links LG'm,"(L, to,t) are two variable quantum invariantsobtained by the Reshetikhin-Turaev construction applied to Hopf superalgebras Uogl(mn). Theseinvariants are known to be generalizations of the Alexander invariant.

Moreover, the Links-Gould invariants seem to inherit some of the properties the Alexander invariant hasdue to its classical homological nature, suggesting we should be able to understand LG as classical linkinvariants.

Using representation theory of Uagl(2l1), we proved in recent work with Guillaume Tahar that thedegree of the Links-Gould polynomial LG?,1 provides a lower bound on the Seifert genus of any knottherefore improving the bound known as the Seifert ineguality in the case of the Alexander invariant. Wealso managed to write LG”,I as a determinant closely related to the Alexander invariant.

Speaker Intro

I was born in France and grew up between France and the United States. l studied math at ENS deCachan (Paris), Université Paris 7 and Université de Bourgogne, where l obtained my PhD (directed byP. Schauenburg and E. Wagner). My mathematical interests are related to low dimensional topology.have been studying knot and link theory, and more precisely connections that exist between classicaland quantum link invariants. On a more personal level, l enjoy spending time with my three childrenCome, Aliocha and Madeleine.